Question

Rewrite the function by completing the square.
\[f(x)=x^{2}-12x-29\]
\[f(x)=(x+\]

\[)^2+\]

Answer 1

To complete the square for the function \( f(x) = x^2 - 12x - 29 \), follow these steps:

1. **Take the linear term (-12) and halve it**:
\[
-\frac{12}{2} = -6
\]

2. **Square this result**:
\[
(-6)^2 = 36
\]

3. **Add and subtract this square inside the function**:
\[
f(x) = (x^2 - 12x + 36) - 36 - 29
\]

4. **Rewrite the quadratic part as a perfect square**:
\[
f(x) = (x - 6)^2 - 36 - 29
\]

5. **Combine the constants**:
\[
-36 - 29 = -65
\]

Putting it all together, we have:
\[
f(x) = (x - 6)^2 - 65
\]

Thus, the completed square form is:
\[
f(x) = (x - 6)^2 - 65
\]

So, you can fill the blanks as follows:
\[ f(x) = (x - 6)^2 - 65 \]
Thus:
\[ f(x) = (x + \mathbf{(-6)})^2 + \mathbf{(-65)} \]